Determining a load status of a platform using a likelihood ratio test

ABSTRACT

In some examples, measurement data is received from at least one sensor that detects a signal reflected from a surface inside a platform. A likelihood ratio test is applied using the measurement data, and a load status of the platform is determined based on the likelihood ratio test.

CROSS REFERENCE TO RELATED APPLICATIONS

This is a continuation of U.S. application Ser. No. 15/350,488, filedNov. 14, 2016, which is hereby incorporated by reference.

BACKGROUND

Trucks, tractor-trailers, or tractors that are connected to chassis forcarrying containers can be used to transport cargo that includes goods.Trucks, tractor-trailers, and containers typically have doors that canbe opened to allow access to cargo that is being transported, and closedto secure the cargo.

BRIEF DESCRIPTION OF THE DRAWINGS

Some implementations of the present disclosure are described withrespect to the following figures.

FIGS. 1A-1B are schematic diagrams of a container hauled by a vehicle,the container including a door and a sensor device according to someimplementations.

FIG. 2 is a block diagram of an example sensor device according to someimplementations.

FIG. 3 is a flow diagram of an example process according to someimplementations.

FIG. 4 a block diagram of an example time-of-flight sensor according tosome implementations.

FIGS. 5A and 5B are graphs plotting output values computed by likelihoodratio tests as a function of measurement samples, according to someimplementations.

FIG. 6 is a graph of an in-phase/quadrature-phase plane showing asubspace used for a likelihood ratio test, according to furtherimplementations.

DETAILED DESCRIPTION

In the present disclosure, use of the term “a,” “an”, or “the” isintended to include the plural forms as well, unless the context clearlyindicates otherwise. Also, the term “includes,” “including,”“comprises,” “comprising,” “have,” or “having” when used in thisdisclosure specifies the presence of the stated elements, but do notpreclude the presence or addition of other elements.

A moveable platform can be used to carry physical items betweendifferent geographic locations. For example, the moveable platform canbe a container (that is attached to a tractor), a truck, or a trailer,where the container provides an enclosed space in which the physicalitems can be stored during shipment. In other examples, the moveableplatform can include another type of carrier structure that is able tocarry physical items. More generally, the moveable platform can be partof, mounted on, or attached, as applicable, to a vehicle, such as atruck, a trailer, a tractor, a car, a train, a ship, an airplane, and soforth. In the ensuing discussion, a moveable platform can be referred toas a container. It is noted that techniques or mechanisms according tosome implementations can be applied to other cargo carrying platformswith an entry barrier that can be opened and closed.

A moveable platform can include a door through which physical items canbe loaded or unloaded into or from an inner chamber of the moveableplatform, respectively. The door is an example of an entry barrier (ormore simply, a “barrier”) that can be opened and closed. Other examplesof barriers include a window or any other structure that can be openedto allow entry through an opening, or closed to block entry through theopening

An entity such as a shipper, a distributor, a manufacturer, a seller ofgoods, or any other entity may wish to track assets (such as cargo) thatare being transported using moveable platforms. To do so, a sensordevice can be mounted on a moveable platform. Sensor devices on variousmoveable platforms can communicate sensor information over a network toa remote service to allow the remote service to track assets that arebeing moved by various moveable platforms. The remote service caninclude a server or a collection of servers and associated networkequipment that may be located at one fixed location or in a mobile unitor as part of a data center or cloud. Asset tracking can involvetracking the current locations of the assets, cargo load status ofmoveable platforms, conditions of the environment around the assets(where such conditions can include a measured temperature, a measuredhumidity, etc.), and/or other information.

A sensor device can include a communication component to communicateover a network. In some examples, sensor devices mounted on moveableplatforms can be part of a larger network of devices. This largernetwork of devices can be part of the “Internet-of-Things” (IoT)technology paradigm to allow different types of devices to communicatedifferent types of data (including sensor data, voice data, video data,e-mail data, picture or image data, messaging data, web browsing data,and so forth). In addition to network traffic communicated by computers,smartphones, wearable devices, and the like, the IoT technology paradigmcontemplates that other types of devices, including householdappliances, vehicles, sensor devices, thermostats, and so forth haveconnectivity to a network to allow the devices to communicate respectivedata.

More generally, asset tracking using network connected sensor devicescan involve acquiring sensor data, transmitting the sensor data, andaggregating such sensor data or producing other measures based on thesensor data to determine information associated with the assets that arebeing transported by moveable platforms. Based on data received from thesensor devices, a server (or servers) can update a database, runanalytics, and/or present status information for display, such thatfurther decisions or actions can be performed. The asset tracking can beused to improve fleet utilization, reduce operating cost, reduce loss ofassets due to theft, and so forth.

In some examples, a measure that can be derived based on measurementsmade by a sensor device mounted on a moveable platform is a load statusof the moveable platform, where the load status can refer to whether ornot the moveable platform contains any physical items such as cargo(i.e., the container is empty or the container contains at least onephysical item). More generally, determining a load status can refer toan amount of loading (e.g., 0% loaded, 10% loaded, 25% loaded, 50%loaded, 100% loaded, or by use of some other measure that can havedifferent values that indicate different levels of loading with physicalitems. Accurately detecting the load status of a container can bechallenging due to one or more factors. For example, the characteristicsof a moveable platform may change over time, such as due to addition ofstructures within the chamber of the moveable platform, presence ofdebris in the moveable platform chamber, cleaning of the inner chamberof the moveable platform, repainting of the inner walls of the moveableplatform, and so forth. In addition, the hardware of the sensor devicecan also vary over time, such as due to varying temperature or due toaging of the hardware, which can affect the sensitivity, gain, and/orother characteristic of a sensor in the sensor device. Moreover,different moveable platforms and different sensor devices may havedifferent characteristics due to variances in manufacturing and/orconfiguration. Furthermore, the size and construction composition of themoveable platform can vary from one moveable platform to another. Also,the surrounding environment for different sensor devices may bedifferent. Due to such varying characteristics of the moveable platformand/or the sensor device, a determination of a load status based onmeasurement data from the sensor device may not be accurate.

Additionally, detection systems for detecting a load status of amoveable platform that rely upon measurements of physical distanceswithin the moveable platform to determine if cargo is present mayproduce sub-optimal results. Sub-optimal detection systems may miss thephysical items within a moveable platform, or may provide a falseindication that the moveable platform is loaded with a physical item.

In accordance with some implementations of the present disclosure,statistical techniques are applied to determine a load status of amoveable platform. In some implementations, the statistical techniquesinclude likelihood ratio test techniques, where a likelihood ratio testgenerally refers to a statistical test that is used to compare thegoodness of fit of two models, including a first model that is a nullmodel and a second model that is an alternative model. The test is basedon a likelihood ratio between the null model representing an emptycontainer and the alternative model representing a loaded container,which expresses how much more likely data fits the null model versus thealternative model. In the context of detecting the load status of amoveable platform, the null model (or null hypothesis) represents amoveable platform is empty, while the alternative model, or alternativehypothesis, represents the moveable platform including at least onephysical item (i.e., the moveable platform is loaded).

In some examples, a likelihood ratio test can include a log likelihoodratio (LLR) test. Another likelihood ratio test is a sequentialprobability ratio test (SPRT), which employs the LLR test to producemultiple LLR outputs for respective multiple measurement data samples.The multiple measurement data samples can include a sequence ofmeasurement data samples collected from at least one sensor fordetecting physical items in a moveable platform. The SPRT technique canaggregate (e.g., sum) the multiple LLR outputs values to produce an SPRTvalue that can be used for determining the load status of a moveableplatform. In further examples, another likelihood ratio test techniquethat can be applied is a general likelihood ratio test (GLRT), which canbe applied in instances where certain information relating to a moveableplatform is missing, such as the color of the moveable platform or otherinformation relating to a characteristic of the moveable platform. TheGLRT technique can thus be applied in cases where partial informationregarding characteristics of the moveable platform are available.

FIGS. 1A-1B illustrate an example truck 100 that includes a tractor unit102 and a container 104 (provided on a chassis) hauled by the tractorunit 102. The container 104 is an example of a moveable platform thatcan be used to carry physical items. FIG. 1B is a perspective view ofthe container 104, to show that the container 104 has a door 106 (orother entry barrier) that is pivotable between an open position and aclosed position (or more generally, moveable between an open positionand a closed position).

In FIG. 1B, the door 106 is in the open position. The door 106 ispivotably mounted to hinges 108, which are attached to a door frame 110of the container 104. The door 106 is able to rotate about the hinges108 between the open position and the closed position. Although FIG. 1Bshows two hinges 108, it is noted that in other examples, the door 106can be mounted on just one hinge, or more than two hinges.

In some examples, a sensor device 112 is mounted to an interior surfaceof the door 106, such that when the door 106 is closed, the sensor 112faces towards an inner chamber 114 of the container 104. As noted above,the container 104 is an example of a moveable platform. The innerchamber 114 of the container 104 (referred to as a “container chamber”in the ensuing discussion) is an example of an inner space of themoveable platform in which physical items can be carried. The containerchamber 114 is defined by the walls of the container 104. In otherexamples, the sensor device 112 is not mounted on the door 106, butrather can be mounted on another surface inside the container 104.

In some implementations, in response to a door-closing event, the sensordevice 112 can make a measurement for use in determining the load statusof the container 104, e.g., to determine whether there is any physicalitem located within the container chamber 114 of the container 104, orto determine a level of loading of physical items within the containerchamber 114. For example, the sensor device 112 includes aTime-of-Flight (ToF) sensor 116.

In some examples, the determination of the load status of the container104 can be performed by a processor (or multiple processors) in thesensor device 112. In other examples, the determination of the loadstatus of the container 104 can be performed by a remote server computer(or multiple remote server computers) that receive the measurement datafrom the sensor device 112.

In FIG. 1A, it is assumed that physical items 118 are located within thecontainer chamber 114. The ToF sensor 116 is able to emit a signal 113that is reflected from a surface inside the container chamber 114. Basedon the detection of the reflected signal by the ToF sensor 116, thesensor device 112 outputs measurement data that can be used inlikelihood ratio test techniques to determine the load status of thecontainer 104.

In examples where the ToF sensor 116 includes a light emitter, such as alight emitting diode (LED), a transmitted periodic signal can includemultiple light pulses that employ an on-off-keying (OOK) of the lightemitter. In other examples, other types of transmitted periodic signalscan be employed. The ToF sensor 116 includes a light detector to detectlight reflected by the light emitter.

In some examples, a light source such as the light emitter mentionedabove may be modulated by a modulation signal. The frequency of themodulation signal can be used in converting the phase shift of thereflected light to distance measure. In some examples, the modulationfrequency can be in a range between 2 megahertz (MHz) to or 5 MHz tomore than 10 MHz, depending upon the ranges of distance to be detected.

In the FIG. 1A example, the emitted signal 113 is reflected from arear-facing surface 120 of the rear-most physical item 118. Therear-facing surface 120 faces the door 106 of the container 104. Inexamples where there are multiple physical items, the ToF sensor 116detects a reflected signal from the physical item 118 that is closest tothe door 106.

If the physical items 118 are not present in the container chamber 114,then the ToF sensor 116 detects a signal reflected from an inner frontsurface 122 of the container 104.

The inner front surface 122 of the container 104 can be the front-mostwall of the container 104. In other examples, another structure such asa frame can be located in the container chamber 114 against whichphysical items 118 can be placed. In such other examples, the innerfront surface 122 is a surface of such other structure, or the floor ofthe container 104.

Although the present discussion refers to examples where the door 106 islocated at the rear of the container 104 and physical items 118 areloaded from the front of the container chamber 114 to the rear of thecontainer chamber 114, it is noted that in other examples, the door 106can be located at a different part of the container 104, such as on aside wall of the container 104, or on a front wall of the container 104.In such other examples, the loading of physical items can be from sideto side, or from the rear to the front. The sensor device 112 can alsobe located on a side wall or the front wall of the container 104 in suchother examples.

In the foregoing discussion, it is assumed that measurement data fromthe sensor device 112 is used to determine the load status of theentirety of the container chamber 114. In other examples, the containerchamber 114 can be divided into multiple zones, and the sensor device112 (or multiple sensor devices 112) can be used to determine the loadstatus of each zone of these multiple zones. Thus, the sensor device(s)112 can determine a load status of a first zone, a load status of asecond zone, and so forth.

FIG. 2 is a block diagram of the sensor device 112 according to someexamples. The sensor device 112 includes the ToF sensor 116, a processor(or multiple processors) 202, and a communication component 204. Aprocessor can include a hardware processing circuit, such as amicroprocessor, a core of a multi-core microprocessor, amicrocontroller, a programmable gate array, a programmable integratedcircuit device, or any other hardware processing circuit.

The sensor device 112 further includes a battery 210 that provides powerto components of the sensor device 112.

In some examples, the processor 202 receives measurement data from theToF sensor 116, and based on the measurement data, the processor 202 isable to determine a load status of the container 104 (either the loadstatus of the entirety of the container, or the load status of a zone ofmultiple zones in the container). The processor 202 provides the loadstatus indication to the communication component 204, which transmitsthe load status indication 206 over a network to a destination, such asa remote service that is used to perform asset tracking.

The communication component 204 can include a wireless transceiver andassociated circuits to allow for wireless communication of output data206 by the sensor device 112 to the network. The wireless communicationcan include wireless communication over a cellular access network, awireless local area network, a satellite network, and so forth.Alternatively, the communication component 204 can include a wiredtransceiver and associated circuits to perform wired communicationsbetween the sensor device 108 and the destination.

In other examples, the sensor device 112 does not perform the loadstatus determination, but rather, the sensor device 112 transmitsmeasurement data based on ToF data of the ToF sensor 116 using thecommunication component 204 to the destination. In such examples, thedetermination of the load status can be performed by a server (orservers) at the destination, based on the measurement data transmittedby the sensor device 112. In such examples, the processor 202 can simplytransfer measurement data received from the ToF sensor 116 to thecommunication component 204 for transmission to the destination.

In examples where the processor 202 of the sensor device 112 isconfigured to perform the load status determination, the processor 202can execute a load status detector 208, which includes software or othermachine-readable instructions executable on the processor 202. In otherexamples, the load status detector 208 can be implemented usinghardware. The load status detector 208 determines, based on measurementdata from the ToF sensor 116, the load status of the container 104.

In some examples, the load status detector 208 includes a likelihoodratio test (LRT) function 209 that is used to apply a likelihood ratiotest using measurement data from the ToF sensor 116 for determining theload status of the container 104. The LRT function 209 can be invoked bythe load status detector 208 to apply any of various likelihood ratiotest techniques according to some examples, such as the LLR testtechnique, SPRT technique, or GLRT technique, which are described ingreater detail below.

Although the LRT function 209 is shown as being part of the load statusdetector 208 in FIG. 2, it is noted that in other examples, the LRTfunction 209 can be separate from the load status detector 208.

In examples where the sensor device 112 does not perform the load statusdetermination, the load status detector 208 and LRT function 209 can beprovided at a server (or multiple servers) that is (are) remote from thesensor device 112. Shifting the determination of the load status of acontainer to a remote server(s), instead of performing the load statusdetermination at the sensor device 112, can reduce the complexity of thesensor device 112, and can allow the sensor device 112 to consume lesspower to save battery power in the sensor device 112.

In some examples, to reduce power consumption of the sensor device 112,and thus to conserve the battery power, the sensor device 112 can bemaintained in a sleep state until an event triggers the sensor device112 to make a measurement and/or to perform processing tasks. A sleepstate refers to a state of the sensor device 112 where the sensor deviceis powered off, or a portion of the sensor device 112 is powered off,such that the sensor device 112 consumes a lower amount of power thananother state of the sensor device, such as an operational state. Anoperational state of the sensor device 112 is a state of the sensordevice 112 where the sensor device is able to perform specified tasks,including measurement of data and/or processing of data. In theoperational state, the sensor device 112 consumes more power than thepower consumed by the sensor device in the sleep state.

In some examples, an event that can trigger the sensor device 112 totransition from the sleep state to the operational state can be a doorclose event, which is generated when the door 106 is closed from an openposition. The sensor device 112 can include a door status sensor 212 todetect either the opening or closing of the door 106 (FIG. 1) or otherentry barrier. Although FIG. 2 shows the door status sensor 212 as beingpart of the sensor device 112, it is noted that in other examples, thedoor status sensor 212 can be external of the sensor device 112.

The door status sensor 212 can detect a change in status of the door 106using any of various mechanisms. For example, a switch can be attachedto the door, where the switch changes state in response to the doorbeing opened or closed. As another example, a magnetic sensor can beused, where the magnetic sensor can be in proximity to a magnet when thedoor is closed, but when the door is opened, the magnetic sensor movesaway from the magnet. The magnetic sensor can thus output differentvalues depending upon whether the door is opened or closed. In otherexamples, acceleration data from an accelerometer and rotation data froma rotation sensor (such as a gyroscope or rotation vector sensor) can beused for detecting the door being opened and closed.

In response to the door status sensor 212 indicating that the door hasbeen closed from an open position, the sensor device 112 can be awakenedfrom the sleep state to the operational state. For example, the ToFsensor 116 can be activated from a lower power state to a higher powerstate, and/or the processor 202 can be activated from a lower powerstate to a higher power state. In some examples, in response todetecting that the door has been closed, the sensor device 112 can waita specified time duration before transitioning from the sleep state tothe operational state, such as to avoid triggering multiple transitionsbetween the sleep state and the operational state in a short period oftime, such as due to a person opening and closing the door in quicksuccession. Additionally, the processor 202 can wait to transition fromthe lower power state to the higher power state to allow the ToF sensor116 time to power on and initialize and take measurements.

FIG. 3 is a flow diagram of an example process for deciding whether amoveable platform such as the container 104 contains a physical item (orphysical items) such as cargo. The process of FIG. 3 can be performed bythe processor 202 in the sensor device 112, or alternatively, by aremote server.

The process of FIG. 3 includes receiving (at 302) measurement data fromat least one sensor (e.g., the ToF sensor 116) that detects a signalreflected from a surface inside the moveable platform. The process ofFIG. 3 further includes applying (at 304) a likelihood ratio test usingthe measurement data. The process determines (at 306) a load status ofthe moveable platform based on the likelihood ratio test.

The likelihood ratio test that is applied using the measurement data canproduce a computed value, which can be compared to one or morethresholds for determining the load status of the moveable platform. Forexample, if the computed value has a first relationship with respect tothe one or more thresholds, then the moveable platform is indicated asbeing empty. On the other hand, if the computed value has a secondrelationship with respect to the one or more thresholds, then themoveable platform is indicated as being loaded. In some examples, thefirst relationship can be a “less than” or “less than or equal”relationship, while the second relationship can be a “greater than” or“greater than or equal” relationship. In other examples, otherrelationships can be employed.

In some examples, the measurement data from the at least one sensor canbe used as a representation of a loaded platform, and the computed valuefrom the likelihood ratio test can be compared with respect to at leastone threshold. Determining the load status of the platform is based onthe comparing.

In some examples, multiple thresholds can be used, including a lowerthreshold and an upper threshold. The computed value calculated based onapplying the likelihood ratio test on measurement data if less than thelower threshold provides an indication that the moveable platform isempty, while the computed value being greater than the upper thresholdprovides an indication that the moveable platform is loaded with atleast one physical item. In other examples, just one threshold can beused.

FIG. 4 is a block diagram of an example ToF sensor 116, which includes alight emitter 402 and a light detector 404. Although just one lightemitter 402 and one light detector 404 are shown in FIG. 4, the ToFsensor 116 in other examples can include multiple light emitters and/ormultiple light detectors. The light emitter 402 can include a lightemitting diode (LED) or another type of light source that can producelight either in the visible spectrum or invisible spectrum (e.g.,infrared or ultraviolet light). The light detector 404 can include aphoto-sensitive diode or other type of light detector.

The light emitter 402 can be used to transmit an emitted light signal406, which can include one or more light pulses in some examples, intothe container chamber 114. An input voltage 401 is applied to the lightemitter 402 to cause the light emitter 402 to transmit the emitted lightsignal 406. The emitted light signal 406 is reflected as a reflectedlight signal 408 from a surface inside the container chamber 104, wherethe surface can be the rear-facing surface 120 of a physical item 118,or the front surface 122 of the container 104 (FIG. 1). The reflectedlight signal 408 is detected by the light detector 404, which producesan output 410 that is responsive to the reflected light signal 408. Theoutput 410 can include an output voltage or an output current, which hasa value (e.g. an amplitude) that is based on a property of the reflectedlight signal 408. Although not shown, the output 410 can be providedthrough a signal chain including intermediate circuits, such as ananalog-to-digital (ADC) converter, an amplifier, a buffer, and so forth.

In other examples, the ToF sensor 116 can include a different type ofsignal emitter to emit another type of signal, such as an acousticsignal, electromagnetic signal, and so forth. In such other examples,the ToF sensor 116 can also include a different type of signal detector,such as an acoustic sensor, an electromagnetic sensor, and so forth.

The following provides further details associated with variouslikelihood ratio tests can be employed for determining a load status ofa moveable platform.

In the ensuing discussion, a signal transmitted by the ToF sensor 116,such as by the light emitter 402 or other signal emitter, can berepresented as x(t). A signal received by the ToF sensor 116, such as bythe light detector 404 or other signal sensor, can be represented asy(t).

In more specific examples, x(t) can be a transmitted signal of N pulsetrains including M pulses per pulse train. Note that in other examples,the transmitted signal x(t) can have a form different from thatdescribed below.

For each pulse train of M pulses, T separates each of the individualpulses of width T/2. Thus a single pulse train of M pulses resembles asquare-wave signal with period T at 50% duty cycle for 0≤t<MT (Mmultiplied by T) and zero for all other times t. By spacing each pulsetrain T_(s) seconds apart (T_(s) represents the length of a samplingperiod), where MT<T_(s), a continuous-time transmitted signal, x(t), canbe expressed as:

$\begin{matrix}{{{x(t)}\overset{\Delta}{=}{\sum\limits_{k = 0}^{N - 1}{\sqrt{P}{p_{T/2}(t)}*{\delta_{T}^{M}\left( {t - {kT}_{s}} \right)}}}},} & \left( {{Eq}.\mspace{14mu} 1} \right)\end{matrix}$where √{square root over (P)} represents a pulse amplitude, and otherparameters of Eq. 1 are provided below. The operator * denotescontinuous-time linear convolution defined asu(t)*v(t)

∫_(−∞) ^(+∞) u(s)v(t−s)ds.  (Eq.2)

Let δ(t) be a Dirac delta function, i.e.,

$\begin{matrix}{{\delta(t)}\overset{\Delta}{=}{\lim\limits_{a->0}{\frac{1}{a\sqrt{\pi}}{e^{{- t^{2}}/a^{2}}.}}}} & \left( {{Eq}.\mspace{14mu} 3} \right)\end{matrix}$

Let δ_(T) ^(M)(t) be a waveform including a train of M Dirac deltafunctions time-shifted and spaced T seconds apart:

$\begin{matrix}{{\delta_{T}^{M}(t)}\overset{\Delta}{=}{\sum\limits_{m = 0}^{M - 1}{{\delta\left( {t - {mT}} \right)}.}}} & \left( {{Eq}.\mspace{14mu} 4} \right)\end{matrix}$

Let p_(w)(t) be a single rectangular pulse of amplitude 1, width W,starting at t=0, ending at t=W, and amplitude zero otherwise, i.e.,

$\begin{matrix}{{p_{W}(t)}\overset{\Delta}{=}\left\{ {\begin{matrix}{1;} & {0 \leq t < W} \\{0;} & {otherwise}\end{matrix}.} \right.} & \left( {{Eq}.\mspace{11mu} 5} \right)\end{matrix}$

As noted above, the likelihood ratio test compares the goodness or fitof two models, a null model (or null hypothesis) and an alternativemodel (or alternative hypothesis).

Let H₀ be the null hypothesis representing an empty container, i.e., nophysical item. For H₀, the received signal y₀(t) is expressed as:y ₀(t)=∝_(c) A _(c) x(t−T _(c))+η(t),  (Eq. 6)where ∝_(c) is the reflection coefficient of the container, A_(c) is theeffective reflective area of the sensor device's field of view cone foran empty container, T_(c) is the round-trip time shift attributed to thecontainer's reflection, and η(t) is additive noise.

Let H₁ be the alternative hypothesis representing at least one physicalitem being present within the container. For the alternative hypothesis,the received signal y₁(t) is expressed as:y ₁(t)=∝_(b) A _(b) x(t−T _(b))+∝_(c)(A _(c) −A _(b))x(t−T_(c))+η(t),  (Eq. 7)where ∝_(b) be the reflection coefficient of the physical item, A_(b) isthe effective reflective area of the physical item, and T_(b) is theround-trip time shift attributed by the physical item's reflection.

In standard form, the hypothesis test isH ₀ : y(t)=y ₀(t) (Empty)H ₁ : y(t)=y ₁(t) (Not Empty),  (Eq. 8)where y(t) is the received signal corresponding to the transmitted x(t),y₀(t) is the received signal corresponding to an empty container, andy₁(t) is the received signal corresponding to a container that is notempty, i.e., a physical item is present. As written, the decision isbinary where 0 means empty and 1 means not empty (i.e., no rejectionregion as written). In other examples, the hypothesis test can begeneralized to indicate more than two different levels of loading of thecontainer.

In some examples, before applying the hypothesis test noted above, areceiver can first multiply a received signal y(t) by in-phase andquadrature-phase (I/Q) square-waveforms, and integrate before samplingusing an analog-to-digital converter (ADC).y _(I)(t)=y(t){tilde over (x)}(t) (In-Phase),  (Eq. 9)y _(Q)(t)=y(t){tilde over (x)}(t−T/4) (Quadrature).  (Eq. 10)

Due to a common DC component, the pair x(t) and x(t−T/4) do not form anorthogonal basis and adds unnecessary mathematical explanation burden tothe phase interpretation. For a more pseudo-intuitive discussion aboutphase using coordinates on an I/Q plot, the DC components and nuisancescalar can be removed as follows:

$\begin{matrix}{\begin{matrix}{{\overset{\sim}{x}(t)}\overset{\Delta}{=}{{\frac{2}{\sqrt{P}}{x(t)}} - 1}} \\{= {{2{\sum\limits_{k = 0}^{N - 1}{{p_{T/2}(t)}*{\delta_{T}^{M}\left( {t - {kT}_{s}} \right)}}}} - 1}}\end{matrix}\begin{matrix}{{\overset{\sim}{x}\left( {t - {T/4}} \right)}\overset{\Delta}{=}{{\frac{2}{\sqrt{P}}{x\left( {t - {T/4}} \right)}} - 1}} \\{= {{2{\sum\limits_{k = 0}^{N - 1}{{p_{T/2}\left( {t - {T/4}} \right)}*{\delta_{T}^{M}\left( {t - {kT}_{s}} \right)}}}} - 1}}\end{matrix}} & \left( {{Eq}.\mspace{14mu} 11} \right)\end{matrix}$According to Eq. 11, an orthogonal basis can be provided in the form ofzero-mean square-waves emulating the sine and cosine waveforms. Notealso that {tilde over (x)}(t) and {tilde over (x)}(t−T/4) still have a50% duty-cycle but now they are antipodal between 1 and −1 amplitudewith zero mean over integer multiples of T.

Note that scaling both {tilde over (x)}(t) and {tilde over (x)}(t−T/4)by 1/√{square root over (MT)} produces an orthonormal basis, {tilde over(x)}(t)/√{square root over (MT)} and {tilde over (x)}(t−T/4√{square rootover (MT)}.

In other examples, an orthogonal basis exists for when the transmittedwaveform resembles a sinusoidal waveform. This may be intentional andmay be done by filtering x(t) with a low-pass filter with a cutofffrequency above 1/T and below an integer number of harmonics, or it maybe unintentional due to the bandwidth restrictions of hardware behavingas a low-pass filter. In these examples, x(t) and x(t−T/4) aresinusoidal waveforms 90° out of phase (e.g., cosine and sine waveformswith a frequency of 1/T hertz) and form a basis corresponding to the I/Qcoefficients.

Next, the receiver then integrates the In-phase & Quadrature (I/Q)signals over NT for each of the N pulse trains comprising of M pulses,i.e., the discrete I/Q signals after integration and sampling can bewritten asy _(I)[n]=∫_(nT) _(s) ^(nT) ^(s) ^(+MT) y(t){tilde over (x)}(t)dt, forn=0,1, . . . ,(N−1) (In-Phase)  (Eq. 11)y _(Q)[n]=∫_(nT) _(s) ^(nT) ^(s) ^(+MT) y(t){tilde over (x)}(t−T/4)dt,for n=0,1, . . . ,(N−1) (Quadrature)  (Eq. 12)

With the following assumptions, the null hypothesis and alternativehypothesis can be expressed according to Eqs. 13-16 further below.

In some examples, the assumptions are as follows:

1) A physical item should be closer than the farthest wall of thecontainer and is within sensor range, i.e. 0≤T_(b)<T_(c)<T/2.

2) The integration time is less than the sampling period, i.e.,0<MT<T_(s).

3) The reflection coefficients are bounded between 0 (black) and 1(white), i.e., 0≤∝_(c)≤1 and 0≤∝_(b)≤1.

4) The reflective area of the physical item is bounded by the area ofthe container, i.e., 0≤A_(b)≤A_(c).

5) The time shift attributed to the physical item is bounded by theempty container, i.e., 0≤T_(b)≤T_(c).

6) Noise sampling is performed as follows:

${{\eta_{I}\lbrack n\rbrack}\overset{\Delta}{=}{\int_{{nT}_{s}}^{{nT}_{S} + {MT}}{{\eta(t)}{\overset{\sim}{x}(t)}{dt}}}},{{\eta_{Q}\lbrack n\rbrack}\overset{\Delta}{=}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(t)}{\overset{\sim}{x}\left( {t - \frac{T}{4}} \right)}{{dt}.}}}}$

7) Let μ_(η)

E{η(t)}, then

$\begin{matrix}\begin{matrix}{\mu_{\eta_{I}}\overset{\Delta}{=}{E\left\{ {\eta_{I}\lbrack n\rbrack} \right\}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{E\left\{ {\eta(t)} \right\}{\overset{\sim}{x}(t)}{dt}}}} \\{= {\mu_{\eta}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\overset{\sim}{x}(t)}{dt}}}}} \\{{= 0},}\end{matrix} & \; \\\begin{matrix}{\mu_{\eta_{Q}} = {E\left\{ {\eta_{Q}\lbrack n\rbrack} \right\}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{E\left\{ {\eta(t)} \right\}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}}} \\{= {\mu_{\eta}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}}}} \\{= 0}\end{matrix} & \; \\{{{\left. 8 \right)\mspace{14mu}{Let}\mspace{14mu}\sigma_{\eta}^{2}}\overset{\Delta}{=}{{E\left\{ \left( {{\eta(t)} - \mu_{\eta}} \right)^{2} \right\}} = {{E\left\{ \left( {\eta(t)} \right)^{2} \right\}} - \mu_{\eta}^{2}}}},{then}} & \; \\\begin{matrix}{\sigma_{\eta_{I}}^{2}\overset{\Delta}{=}{E\left\{ \left( {\eta_{I}\lbrack n\rbrack} \right)^{2} \right\}}} \\{= {E\left\{ {\left( {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(t)}{\overset{\sim}{x}(t)}{dt}}} \right)\left( {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(s)}{\overset{\sim}{x}(s)}{ds}}} \right)} \right\}}} \\{= {E\left\{ {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(t)}{\eta(s)}{\overset{\sim}{x}(t)}{\overset{\sim}{x}(s)}{dsdt}}}} \right\}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{E\left\{ {{\eta(t)}{\eta(s)}} \right\}{\overset{\sim}{x}(t)}{\overset{\sim}{x}(s)}{dsdt}}}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\delta\left( {s - t} \right)}\left( {\sigma_{\eta}^{2} + \mu_{\eta}^{2}} \right){\overset{\sim}{x}(t)}{\overset{\sim}{x}(s)}{dsdt}}}}} \\{= {\left( {\sigma_{\eta}^{2} + \mu_{\eta}^{2}} \right){\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\left( {\overset{\sim}{x}(t)} \right)^{2}{dt}}}}} \\{{= {\left( {\sigma_{\eta}^{2} + \mu_{\eta}^{2}} \right){MT}}},}\end{matrix} & \; \\\begin{matrix}{\sigma_{\eta_{Q}}^{2}\overset{\Delta}{=}{E\left\{ \left( {\eta_{Q}\lbrack n\rbrack} \right)^{2} \right\}}} \\{= {E\left\{ {\left( {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(t)}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}} \right)\left( {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(s)}{\overset{\sim}{x}\left( {s - {T/4}} \right)}{ds}}} \right)} \right\}}} \\{= {E\left\{ {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(t)}{\eta(s)}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{\overset{\sim}{x}\left( {s - {T/4}} \right)}{dsdt}}}} \right\}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{E\left\{ {{\eta(t)}{\eta(s)}} \right\}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{\overset{\sim}{x}\left( {s - {T/4}} \right)}{dsdt}}}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\delta\left( {s - t} \right)}\left( {\sigma_{\eta}^{2} + \mu_{\eta}^{2}} \right){\overset{\sim}{x}\left( {t - {T/4}} \right)}{\overset{\sim}{x}\left( {s - {T/4}} \right)}{dsdt}}}}} \\{= {\left( {\sigma_{\eta}^{2} + \mu_{\eta}^{2}} \right){\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\left( {\overset{\sim}{x}\left( {t - {T/4}} \right)} \right)^{2}{dt}}}}} \\{= {\left( {\sigma_{\eta}^{2} + \mu_{\eta}^{2}} \right){MT}}}\end{matrix} & \;\end{matrix}$

According to the foregoing assumptions, the null hypothesis (H₀: EMPTY)in which I/Q discrete sampling has been applied is expressed as Eqs. 13and 14 below.

$\begin{matrix}\begin{matrix}{{y_{I|0}\lbrack n\rbrack} = {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{y_{0}(t)}{\overset{\sim}{x}(t)}{dt}}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\left( {\propto_{c}{{A_{c}{x\left( {t - T_{c}} \right)}} + {\eta(t)}}} \right){\overset{\sim}{x}(t)}{dt}}}} \\{= {\propto_{c}{{A_{c}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{x\left( {t - T_{c}} \right)}{\overset{\sim}{x}(t)}{dt}}}} + {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(t)}{\overset{\sim}{x}(t)}{dt}}}}}} \\{= {{2\sqrt{P}} \propto_{c}{A_{c}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}\left( {{p_{T/2}\left( {t - T_{c}} \right)}*{\delta_{T}^{M}\left( {t - {nT}_{s}} \right)}} \right)}}}} \\{{{\left( {{p_{T/2}(t)}*{\delta_{T}^{M}\left( {t - {nT}_{s}} \right)}} \right){dt}} - \sqrt{P}} \propto_{c}{A_{c}\int_{{nT}_{s}}^{{nT}_{s} + {MT}}}} \\{{{p_{T/2}\left( {t - T_{c}} \right)}*{\delta_{T}^{M}\left( {t - {nT}_{s}} \right)}{dt}} + {\eta_{I}\lbrack n\rbrack}} \\{= {{2\sqrt{P}} \propto_{c}{{A_{c}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\left( {{p_{{T/2} - T_{c}}\left( {t - T_{c}} \right)}*{\delta_{T}^{M}\left( {t - {nT}_{s}} \right)}} \right){dt}}}} -}}} \\{\sqrt{P} \propto_{c}{{{A_{c}\left( {T/2} \right)}M} + {\eta_{I}\lbrack n\rbrack}}} \\{= {{2\sqrt{P}} \propto_{c}{{{A_{c}\left( {{T/2} - T_{c}} \right)}M} - \sqrt{P}} \propto_{c}{{{A_{c}\left( {T/2} \right)}M} + {\eta_{I}\lbrack n\rbrack}}}} \\{= {{2\sqrt{P}} \propto_{c}{{{A_{c}\left( {T/2} \right)}M} - {2\sqrt{P}}} \propto_{c}{{A_{c}T_{c}M} -}}} \\{\sqrt{P} \propto_{c}{{{A_{c}\left( {T/2} \right)}M} + {\eta_{I}\lbrack n\rbrack}}} \\{{= {\propto_{c}{{A_{c}M\sqrt{P}\left( {{T/2} - {2T_{c}}} \right)} + {\eta_{I}\lbrack n\rbrack}}}},}\end{matrix} & \left( {{Eq}.\mspace{14mu} 13} \right) \\\begin{matrix}{{y_{Q|0}\lbrack n\rbrack} = {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{y_{0}(t)}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\left( {\propto_{c}{{A_{c}{x\left( {t - T_{c}} \right)}} + {\eta(t)}}} \right){\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}}} \\{= {\propto_{c}{{A_{c}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{x\left( {t - T_{c}} \right)}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}}} +}}} \\{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(t)}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}} \\{= {{2\sqrt{P}} \propto_{c}{A_{c}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}\left( {{p_{T/2}\left( {t - T_{c}} \right)}*{\delta_{T}^{M}\left( {t - {nT}_{s}} \right)}} \right)}}}} \\{{{\left( {{p_{T/2}\left( {t - {T/4}} \right)}*{\delta_{T}^{M}\left( {t - {nT}_{s}} \right)}} \right){dt}} - \sqrt{P}} \propto_{c}{A_{c}\int_{{nT}_{s}}^{{nT}_{s} + {MT}}}} \\{{{p_{T/2}\left( {t - T_{c}} \right)}*{\delta_{T}^{M}\left( {t - {nT}_{s}} \right)}{dt}} + {\eta_{Q}\lbrack n\rbrack}} \\{= {{2\sqrt{P}} \propto_{c}{{A_{c}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\begin{pmatrix}{{p_{{T/2} - {{{T/4} - T_{c}}}}\left( {t - T_{c}} \right)}*} \\{\delta_{T}^{M}\left( {t - {nT}_{s}} \right)}\end{pmatrix}{dt}}}} -}}} \\{\sqrt{P} \propto_{c}{{{A_{c}\left( {T/2} \right)}M} + {\eta_{Q}\lbrack n\rbrack}}} \\{= {{2\sqrt{P}} \propto_{c}{{{A_{c}\left( {{T/2} - {{{T/4} - T_{c}}}} \right)}M} -}}} \\{\sqrt{P} \propto_{c}{{{A_{c}\left( {T/2} \right)}M} + {\eta_{Q}\lbrack n\rbrack}}} \\{= {{2\sqrt{P}} \propto_{c}{{{A_{c}\left( {T/2} \right)}M} - {2\sqrt{P}}} \propto_{c}{{A_{c}{{{T/4} - T_{c}}}M} -}}} \\{\sqrt{P} \propto_{c}{{{A_{c}\left( {T/2} \right)}M} + {\eta_{Q}\lbrack n\rbrack}}} \\{= {\propto_{c}{{A_{c}M\sqrt{P}\left( {{T/2} - {2{{{T/4} - T_{c}}}}} \right)} + {{\eta_{Q}\lbrack n\rbrack}.}}}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 14} \right)\end{matrix}$

According to the foregoing assumptions, the alternative hypothesis (H₁:NOT EMPTY) in which I/Q discrete sampling has been applied is expressedas Eqs. 15 and 16 below.

$\begin{matrix}\begin{matrix}{{y_{I|1}\lbrack n\rbrack} = {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{y_{1}(t)}{\overset{\sim}{x}(t)}{dt}}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\begin{pmatrix}{\propto_{b}{{A_{b}{x\left( {t - T_{b}} \right)}} +}} \\{\propto_{c}{{\left( {A_{c} - A_{b}} \right)x\left( {t - T_{c}} \right)} + {\eta(t)}}}\end{pmatrix}{\overset{\sim}{x}(t)}{dt}}}} \\{= {\propto_{b}{{A_{b}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{x\left( {t - T_{b}} \right)}{\overset{\sim}{x}(t)}{dt}}}} +} \propto_{c}\left( {A_{c} - A_{b}} \right)}} \\{{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{x\left( {t - T_{c}} \right)}{\overset{\sim}{x}(t)}{dt}}} + {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(t)}{\overset{\sim}{x}(t)}{dt}}}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 15} \right) \\\begin{matrix}{\mspace{70mu}{= {\propto_{b}{{A_{b}M\sqrt{P}\left( {{T/2} - {2T_{b}}} \right)} +} \propto_{c}{{A_{c}M\sqrt{P}\left( {{T/2} - {2T_{c}}} \right)} -}}}} \\{\propto_{c}{{A_{b}M\sqrt{P}\left( {{T/2} - {2T_{c}}} \right)} + {\eta_{I}\lbrack n\rbrack}}} \\{= {\propto_{c}{{A_{b}M\sqrt{P}\left( {{T/2} - {2T_{b}}} \right)} -} \propto_{c}{{A_{b}M\sqrt{P}\left( {{T/2} - {2T_{c}}} \right)} +}}} \\{y_{I|0}\lbrack n\rbrack} \\{{= {{A_{b}M\sqrt{P}\left( {\propto_{b}{\left( {{T/2} - {2T_{b}}} \right) -} \propto_{c}\left( {{T/2} - {2T_{c}}} \right)} \right)} + {y_{I|0}\lbrack n\rbrack}}},}\end{matrix} & \; \\\begin{matrix}{{y_{Q|1}\lbrack n\rbrack} = {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{y_{1}(t)}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}}} \\{= {\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{\begin{pmatrix}{\propto_{c}{{A_{c}{x\left( {t - T_{b}} \right)}} +}} \\{\propto_{c}{{\left( {A_{c} - A_{b}} \right)x\left( {t - T_{c}} \right)} + {\eta(t)}}}\end{pmatrix}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}}} \\{= {\propto_{b}{{A_{b}{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{x\left( {t - T_{b}} \right)}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}}} +} \propto_{c}\left( {A_{c} - A_{b}} \right)}} \\{{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{x\left( {t - T_{c}} \right)}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}} +} \\{\int_{{nT}_{s}}^{{nT}_{s} + {MT}}{{\eta(t)}{\overset{\sim}{x}\left( {t - {T/4}} \right)}{dt}}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 16} \right) \\{\mspace{65mu}\begin{matrix}{= {\propto_{b}{{A_{b}M\sqrt{P}\left( {{T/2} - {2{{{T/4} - T_{b}}}}} \right)} +}}} \\{\propto_{c}{{A_{c}M\sqrt{P}\left( {{T/2} - {2{{{T/4} - T_{c}}}}} \right)} -}} \\{\propto_{c}{{A_{b}M\sqrt{P}\left( {{T/2} - {2{{{T/4} - T_{c}}}}} \right)} + {\eta_{Q}\lbrack n\rbrack}}} \\{= {\propto_{c}{{A_{b}M\sqrt{P}\left( {{T/2} - {2{{{T/4} - T_{b}}}}} \right)} -}}} \\{\propto_{c}{{A_{b}M\sqrt{P}\left( {{T/2} - {2{{{T/4} - T_{c}}}}} \right)} + {y_{Q|0}\lbrack n\rbrack}}} \\{= {{A_{b}M\sqrt{P}\begin{pmatrix}{\propto_{b}{\left( {{T/2} - {2{{{T/4} - T_{b}}}}} \right) -}} \\{\propto_{c}\left( {{T/2} - {2{{{T/4} - T_{c}}}}} \right)}\end{pmatrix}} + {y_{Q|0}\lbrack n\rbrack}}}\end{matrix}} & \;\end{matrix}$

For readability, the following substitutions are used:U _(b)

T/2−2T _(b) , V _(b)

T/2−2|T/4−T _(b)|; (0≤T _(b) ≤T/2)U _(c)

T/2−2T _(c) , V _(c)

T/2−2|T/4−T _(c)|; (0≤T _(c) ≤T/2).

Thus, the received discrete I&Q samples for both H₀ and H₁ areH ₀ : y _(I|0)[n]=∝_(c) A _(c) M√{square root over (P)}U _(c)+η_(I)[n].y _(Q|0)[n]=∝_(c) A _(c) M√{square root over (P)}V _(c)+η_(Q)[n].  (Eq.17)H ₁ : y _(I|1)[n]=A _(b) M√{square root over (P)}(∝_(b) U _(b)−∝_(c) U_(c))+y _(I|0)[n].y _(Q|1)[n]=A _(b) M√{square root over (P)}(∝_(b) V _(b)−∝_(c) V _(c))+y_(Q|0)[n].  (Eq. 18)

For H₁, both y_(I|1)[n] and y_(Q|1)[n] are in terms of y_(I|0)[n] andy_(Q|0)[n], respectfully, along with corresponding biasing terms forwhen cargo is present. In other words, the above model describes thecase where a physical item is present using terms of when the containeris empty along with the adjustments to account for the physical itembeing present. One interpretation is that the biasing terms provide a“delta offset” to a “baseline” vector.Δ_(I)[n]

A _(b) M√{square root over (P)}(∝_(b) U _(b)−∝_(c) U _(c)).  (Eq. 19)Δ_(Q)[n]

A _(b) M√{square root over (P)}(∝_(b) V _(b)−∝_(c) V _(c)).  (Eq. 20)

In column-vector form, the discrete samples becomeΔ[n]=(Δ_(I)[n]Δ_(Q)[n])′.  (Eq. 21)H ₀ : y ₀[n]=(y _(I|0)[n]y _(Q|0)[n])′.  (Eq. 22)H ₁ : y ₁[n]=(y _(I|1)[n]y _(Q|1)[n])′=Δ[n]+y ₀[n].  (Eq. 23)

Log-Likelihood Ratio (LLR) Test

A likelihood ratio test λ(y) and the log-likelihood ratio (LLR) testLLR(y) can be expressed as follows:

$\begin{matrix}{{\lambda(y)}\overset{\Delta}{=}{{\frac{f\left( y \middle| 1 \right)}{f\left( y \middle| 0 \right)}\underset{0}{\overset{1}{\gtrless}}\left. 1\Longrightarrow{{LLR}(y)} \right.}\overset{\Delta}{=}{{\ln\;{\lambda(y)}}\underset{0}{\overset{1}{\gtrless}}0.}}} & \left( {{Eq}.\mspace{14mu} 24} \right) \\{{{LLR}(y)}\overset{\Delta}{=}{{\ln\;{\lambda(y)}} = {{\ln\;{f\left( y \middle| 1 \right)}} - {\ln\;{{f\left( y \middle| 0 \right)}.}}}}} & \left( {{Eq}.\mspace{14mu} 25} \right)\end{matrix}$

For the equi-variance multi-variable Gaussian noise model, the followingprobability density distributions,

$\begin{matrix}{{{f\left( y \middle| 0 \right)}\overset{\Delta}{=}{\frac{1}{2\pi{\Sigma }}{\exp\left( {{- \frac{1}{2}}\left( {{y -}\mu_{0}} \right)^{\prime}{\Sigma^{- 1}\left( {y - \mu_{0}} \right)}} \right)}}},} & \left( {{Eq}.\mspace{14mu} 26} \right) \\{{{f\left( y \middle| 1 \right)}\overset{\Delta}{=}{\frac{1}{2\pi{\Sigma }}{\exp\left( {{- \frac{1}{2}}\left( {y - \mu_{1}} \right)^{\prime}{\Sigma^{- 1}\left( {y - \mu_{1}} \right)}} \right)}}},} & \left( {{Eq}.\mspace{14mu} 27} \right)\end{matrix}$represent an empty container and a loaded container, respectfully. InEq. 26, the parameter μ₀ represents an estimated mean of the nullhypothesis y₀[n] signals, and in Eq. 27, the parameter μ₁ represents anestimated mean of the alternative hypothesis y₁[n] signals. Note thaty₀[n] represents received signals (such as signals received from thelight detector 404 corresponding to an empty container, and y₁[n]represents received signals corresponding to a non-empty container. Foran example channel model (model of a communications medium through whichlight signals are transmitted and received) and assuming zero-meannoise, the additive noise covariance matrix can be expressed asΣ

E{ηη′}.  (Eq. 28)and the LLR function LLR(y) becomesLLR(y)=½((y−μ ₀)′Σ⁻¹(y−μ ₀)−(y−μ ₁)′Σ⁻¹(y−μ ₁))  (Eq. 29)

If the additive noise is assumed to be additive white Gaussian noise(AWGN), thenΣ=σ_(η) ² I ₂,  (Eq. 30)and the LLR function LLR(y) simplifies further to

$\begin{matrix}\begin{matrix}{{{LLR}(y)} = {\frac{1}{2\sigma_{\eta}^{2}}\left( {{\left( {y - \mu_{0}} \right)^{\prime}\left( {y - \mu_{0}} \right)} - {\left( {y - \mu_{1}} \right)^{\prime}\left( {y - \mu_{1}} \right)}} \right)}} \\{= {\frac{1}{2\sigma_{\eta}^{2}}\left( {{\left( {y - \mu_{0}} \right)^{\prime}\left( {y - \mu_{0}} \right)} + {\left( {\mu_{0} + \Delta - y} \right)^{\prime}\left( {y - \mu_{0} - \Delta} \right)}} \right)}} \\{{= {\frac{1}{2\sigma_{\eta}^{2}}\left( {{2\left( {y^{\prime} - \mu_{0}^{\prime}} \right)\Delta} - {\Delta^{\prime}\Delta}} \right)}},}\end{matrix} & \left( {{Eq}.\mspace{14mu} 31} \right)\end{matrix}$whereμ₁=μ₀+Δ,  (Eq. 32)

Now, the log-likelihood ratio test can be written in terms of a discretesignal

$\begin{matrix}{{{{LLR}\left( {y\lbrack n\rbrack} \right)}\overset{\Delta}{=}{\frac{1}{2\sigma_{\eta}^{2}}\left( {{2\left( {{y^{\prime}\lbrack n\rbrack} - {{\hat{\mu}}_{0}^{\prime}\lbrack n\rbrack}} \right){\hat{\Delta}\lbrack n\rbrack}} - {{{\hat{\Delta}}^{\prime}\lbrack n\rbrack}{\hat{\Delta}\lbrack n\rbrack}}} \right)}},} & \left( {{Eq}.\mspace{14mu} 33} \right)\end{matrix}$where{circumflex over (μ)}₁[n]={circumflex over (μ)}₀[n]+{circumflex over(Δ)}[n],  (Eq. 34){circumflex over (Δ)}[n]={circumflex over (μ)}₁[n]−{circumflex over(μ)}₀[n]  (Eq. 35)

In Eq. 34, {circumflex over (μ)}₀[n] represents an estimated baselinecorresponding to an empty container.

FIG. 5A shows an example depicting values of the LLR function LLR(y[n])calculated as set forth above as data samples y[n] are received, where nis the sample index represented by the horizontal axis in FIG. 5A, andcurve 502 represents the values of LLR(y[n]) at each sample index.

As discussed further above, the output values of the log-likelihoodratio test, LLR(y[n]), can be compared to thresholds (including an upperthreshold and a lower threshold) to determine whether a container isempty or loaded. However, as seen in FIG. 5A, the LLR(y[n]) values varywidely as received signal samples are continually received, and thus maynot accurately determine the load status of the container. As discussedfurther below, the sequential probability ratio test (SPRT) can be usedinstead, which aggregates (e.g., sums), multiple LLR(y[n]) values.

To set the upper and lower thresholds, consider

A < LLR(y[n]) < B$A < {\frac{1}{2\sigma_{\eta}^{2}}\left( {{2\left( {{y^{\prime}\lbrack n\rbrack} - {{\hat{\mu}}_{0}^{\prime}\lbrack n\rbrack}} \right){\hat{\Delta}\lbrack n\rbrack}} - {{{\hat{\Delta}}^{\prime}\lbrack n\rbrack}{\hat{\Delta}\lbrack n\rbrack}}} \right)} < B$σ_(η)²A < (y^(′)[n] − μ̂₀^(′)[n])Δ̂[n] − Δ̂^(′)[n]Δ̂[n]/2 < σ_(η)²B${{\sigma_{\eta}^{2}A} < {\overset{\sim}{S}}_{0} < {\sigma_{\eta}^{2}B}},$where {A, B} ∈

are real-numbered thresholds selected to control the type II and type Ierror rates, respectively. Estimated thresholds are calculated asfollows, where π _(L) represents the lower threshold, and π _(U)represents the upper threshold.π _(L)

{circumflex over (σ)}_(η) ²[n]A, π _(U)

{circumflex over (σ)}_(η) ²[n]B.  (Eq. 36)

More generally, the upper and lower thresholds are computed based ontuning parameters (such as A and B) selected to control error rates.

In practice, obtaining a baseline estimate for {circumflex over (μ)}₁[n]may prove difficult because of load dependency, so the process can startby assuming that the observation y[n] is that of a loaded container,then{circumflex over (Δ)}[n]={circumflex over (μ)}_(y)[n]−{circumflex over(μ)}₀[n],  (Eq. 37)can be used as the estimated “delta offset” where {circumflex over(μ)}_(y)[n] is the estimated sample mean of y[n].

$\begin{matrix}\begin{matrix}{{H_{0}\text{:}E\left\{ {\hat{\Delta}\lbrack n\rbrack} \middle| 0 \right\}} = {E\left\{ {{{\hat{\mu}}_{y}\lbrack n\rbrack} - {{\hat{\mu}}_{0}\lbrack n\rbrack}} \right\}}} \\{= {{E\left\{ {{\hat{\mu}}_{y}\lbrack n\rbrack} \right\}} - {E\left\{ {{\hat{\mu}}_{0}\lbrack n\rbrack} \right\}}}} \\{= {\mu_{0} - \mu_{0}}} \\{{= {\underset{\_}{0}}_{2 \times 1}},.}\end{matrix} & \left( {{Eq}.\mspace{14mu} 38} \right) \\\begin{matrix}{{H_{1}\text{:}E\left\{ {\hat{\Delta}\lbrack n\rbrack} \middle| 1 \right\}} = {E\left\{ {{{\hat{\mu}}_{y}\lbrack n\rbrack} - {{\hat{\mu}}_{0}\lbrack n\rbrack}} \right\}}} \\{= {{E\left\{ {{\hat{\mu}}_{y}\lbrack n\rbrack} \right\}} - {E\left\{ {{\hat{\mu}}_{0}\lbrack n\rbrack} \right\}}}} \\{= {\mu_{1} - \mu_{0}}} \\{= {\Delta.}}\end{matrix} & \;\end{matrix}$

For a single received signal sample, {circumflex over (μ)}_(n)[n]=y[n],and{circumflex over (Δ)}[n]=y[n]−{circumflex over (μ)}₀[n].  (Eq. 39)and the LLR expression simplifies to

$\begin{matrix}{{{LLR}(y)} = {\frac{1}{2\sigma_{\eta}^{2}}{\left( {\Delta_{I}^{2} + \Delta_{Q}^{2}} \right).}}} & \left( {{Eq}.\mspace{14mu} 40} \right)\end{matrix}$

Going back to the case where the noise may be jointly correlated, thelog-likelihood ratio test is expressed as:LLR(y)=½((y−μ ₀)′Σ⁻¹(y−μ ₀)−(y−μ ₁)′Σ⁻¹(y−μ ₁)),  (Eq. 41)for the LLR expression where the noise covariance matrix and itsinverse, respectively, are

$\begin{matrix}{{{\Sigma = \begin{pmatrix}\sigma_{I}^{2} & {\rho\;\sigma_{I}\sigma_{Q}} \\{\rho\;\sigma_{I}\sigma_{Q}} & \sigma_{Q}^{2}\end{pmatrix}};}{\Sigma^{- 1} = {\begin{pmatrix}\sigma_{I}^{- 2} & {{- \rho}\;\sigma_{Q}^{- 1}\sigma_{I}^{- 1}} \\{{- {\rho\sigma}_{I}^{- 1}}\sigma_{Q}^{- 1}} & \sigma_{Q}^{- 2}\end{pmatrix}{\frac{1}{1 - \rho^{2}}.}}}} & \left( {{Eq}.\mspace{14mu} 42} \right)\end{matrix}$

Using again, μ₁=μ₀+Δ and Δ=μ₀−μ₁, the LLR expression simplifies toLLR(y)=(y−μ ₀)′Σ⁻¹Δ−Δ′Σ⁻¹Δ/2.  (Eq. 43)

Again in practice, obtaining a baseline estimate for {circumflex over(μ)}₁[n] may prove difficult because of load dependency, so the processcan start by assuming that the observation y[n] is that of a loadedcontainer, then for a single sample{circumflex over (Δ)}[n]=y[n]−{circumflex over (μ)}₀[n],  (Eq. 44)

Thus, the LLR expression further simplifies to

$\begin{matrix}{{{LLR}(y)} = {\frac{1}{2\left( {1 - \rho^{2}} \right)}{\left( {{\Delta_{I}^{2}\sigma_{I}^{- 2}} + {\Delta_{Q}^{2}\sigma_{Q}^{- 2}} - {2\Delta_{Q}\Delta_{I}{\rho\sigma}_{I}^{- 1}\sigma_{Q}^{- 1}}} \right).}}} & \left( {{Eq}.\mspace{14mu} 45} \right)\end{matrix}$

If σ_(Q)=σ_(I)=σ_(η), then

$\begin{matrix}{{{LLR}(y)} = {\frac{1}{2\left( {1 - \rho^{2}} \right)\sigma_{\eta}^{2}}{\left( {\Delta_{I}^{2} + \Delta_{Q}^{2} - {2\Delta_{Q}\Delta_{I}\rho}} \right).}}} & \left( {{Eq}.\mspace{14mu} 46} \right)\end{matrix}$

If ρ=0, then

$\begin{matrix}{{{{LLR}(y)} = {\frac{1}{2\sigma_{\eta}^{2}}\left( {\Delta_{I}^{2} + \Delta_{Q}^{2}} \right)}},} & \left( {{Eq}.\mspace{14mu} 47} \right)\end{matrix}$which is the same as above when assuming AWGN.

Sequential Probability Ratio Test (SPRT):

The following describes the SPRT in detail, according to some examples.As noted above, the SPRT sums or otherwise aggregates a number of LLRoutput values to produce an SPRT output value, and the SPRT output value(represented by curve 504 in FIG. 5B) can be compared to an upperthreshold 506 and a lower threshold 508 as depicted in FIG. 5B. FIG. 5Bdepicts SPRT output values as a function of received signal samples(sample indexes). Because the SPRT output value is based on the sum of anumber of LLR output values, FIG. 5B shows the SPRT output value 504 ascontinually increasing as additional received signal samples arereceived. When the SPRT output value 504 exceeds the upper threshold506, that indicates that cargo has been detected, and the container canbe marked as loaded. In contrast, when the SPRT output value 504 dropsbelow the lower threshold 506, that indicates that the container isempty.

Generally, in employing the SPRT, multiple values are computed using alikelihood ratio test function (e.g., the LLR function discussed above)applied on corresponding samples of measurement data received from theat least one sensor. The multiple values are aggregated to produce anaggregate value, and then compared to at least one threshold todetermine a load status of a container.

Using the vector notation from earlier (see y₀[n], y₁[n] and Δ[n]), lety[n]

(y _(I)[n]y _(Q)[n]),  (Eq. 48)be the received discrete coordinate samples, wherey _(I)[n]

∫_(nT) _(s) ^(nT) ^(s) ^(+MT) y(t){tilde over (x)}(t)dt, y _(Q)[n]

∫_(nT) _(s) ^(nT) ^(s) ^(+MT) y(t){tilde over (x)}(t−T/4) dt.  (Eq. 49)

Further expanding on this notation,y[0:n]

(y _(I)[0:n]y _(Q)[0:n]),  (Eq. 50)y _(I)[0:n]

(y _(I)[0]y _(I)[1] . . . y _(I)[n])′,  (Eq. 51)y _(Q)[0:n]

(y _(Q)[0]y _(Q)[1] . . . y _(Q)[n])′,  (Eq. 52)with each including the latest (n+1) samples.

$\begin{matrix}{\mspace{20mu}{{{\lambda\left( {y\left\lbrack {0\text{:}n} \right\rbrack} \right)}\overset{\Delta}{=}{\frac{f\left( {y\left\lbrack {0\text{:}n} \right\rbrack} \middle| 1 \right)}{f\left( {y\left\lbrack {0\text{:}n} \right\rbrack} \middle| 0 \right)} = \frac{\prod\limits_{k = 0}^{n}{f\left( {y\lbrack k\rbrack} \middle| 1 \right)}}{\prod\limits_{k = 0}^{n}{f\left( {y\lbrack k\rbrack} \middle| 0 \right)}}}},}} & \left( {{Eq}.\mspace{14mu} 53} \right) \\{{{{LLR}\left( {y\left\lbrack {0\text{:}n} \right\rbrack} \right)}\overset{\Delta}{=}{{\ln\;{\lambda\left( {y\left\lbrack {0\text{:}n} \right\rbrack} \right)}} = {{{\sum\limits_{k = 0}^{n}{f\left( {y\lbrack k\rbrack} \middle| 1 \right)}} - {\sum\limits_{k = 0}^{n}{f\left( {y\lbrack k\rbrack} \middle| 0 \right)}}} = {\sum\limits_{k = 0}^{n}{{LLR}\left( {y\lbrack k\rbrack} \right)}}}}},} & \left( {{Eq}.\mspace{14mu} 54} \right) \\{\mspace{20mu}\begin{matrix}{S_{n}\overset{\Delta}{=}{\sum\limits_{k = 0}^{n}{{LLR}\left( {y\lbrack k\rbrack} \right)}}} \\{= {\sum\limits_{k = 0}^{n}{\frac{1}{2\sigma_{\eta}^{2}}{\left( {{2\left( {{y^{\prime}\lbrack k\rbrack} - {{\hat{\mu}}_{0}^{\prime}\lbrack k\rbrack}} \right){\hat{\Delta}\lbrack k\rbrack}} - {{{\hat{\Delta}}^{\prime}\lbrack k\rbrack}{\hat{\Delta}\lbrack k\rbrack}}} \right).}}}}\end{matrix}} & \left( {{Eq}.\mspace{14mu} 55} \right)\end{matrix}$

In Eq. 55, S_(n) represents the SPRT output value derived from summingn+1 LLR output values.

In order to set the upper and lower thresholds (represented as 506 and508 in FIG. 5B), consider

A < S_(n) < B$A < {\frac{1}{2\sigma_{\eta}^{2}}{\sum\limits_{k = 0}^{n}\left( {{2\left( {{y^{\prime}\lbrack k\rbrack} - {{\hat{\mu}}_{0}^{\prime}\lbrack k\rbrack}} \right){\hat{\Delta}\lbrack k\rbrack}} - {{{\hat{\Delta}}^{\prime}\lbrack k\rbrack}{\hat{\Delta}\lbrack k\rbrack}}} \right)}} < B$${\sigma_{\eta}^{2}A} < {{\sum\limits_{k = 0}^{n}{\left( {{y^{\prime}\lbrack k\rbrack} - {{\hat{\mu}}_{0}^{\prime}\lbrack k\rbrack}} \right){\hat{\Delta}\lbrack k\rbrack}}} - {\sum\limits_{k = 0}^{n}{{{\hat{\Delta}}^{\prime}\lbrack k\rbrack}{{\hat{\Delta}\;\lbrack k\rbrack}/2}}}} < {\sigma_{\eta}^{2}B}$${{\sigma_{\eta}^{2}A} < {\overset{\sim}{S}}_{n} < {\sigma_{\eta}^{2}B}},$where {A, B} ∈

are real-numbered thresholds selected to control the type II and type Ierror rates, respectively, and

$\begin{matrix}{{{\overset{\sim}{S}}_{n}\overset{\Delta}{=}{\sum\limits_{k = 0}^{n}\left( {{\left( {{y^{\prime}\lbrack k\rbrack} - {{\hat{\mu}}_{0}^{\prime}\lbrack k\rbrack}} \right){\hat{\Delta}\lbrack k\rbrack}} - \frac{{{\hat{\Delta}}^{\prime}\lbrack k\rbrack}{\hat{\Delta}\lbrack k\rbrack}}{2}} \right)}},} & \left( {{Eq}.\mspace{14mu} 56} \right)\end{matrix}$

In practice, implementations can use the sampled statistic {circumflexover (σ)}_(η) ²[n] to estimate σ_(η) ² which inherently has a variance,so to mitigate early termination due to small sample size variation, aprocess can “gear-step” the upper and lower thresholds using any numberof schemes, e.g.,

${{\underset{\_}{\pi}}_{L} < {\overset{\sim}{S}}_{n} < {{\overset{\_}{\pi}}_{U} - \infty} < {\overset{\sim}{S}}_{n} < {+ \infty}};{n < N_{\infty}}$${{{{\hat{\sigma}}_{\eta}^{2}\lbrack n\rbrack}\left( {A - {Ce}^{({1 - n})}} \right)} < {\overset{\sim}{S}}_{n} < {{{\hat{\sigma}}_{\eta}^{2}\lbrack n\rbrack}\left( {B + {De}^{({1 - n})}} \right)}};{n \geq N_{\infty}}$where {A, B, C, D} ∈

are real-numbered constants selected to control the type I and type IIerror rates and N_(∞) is the number of samples to wait allowing forsettlement of the sample statistic variances (i.e., minimum number ofobserved samples before allowing detection to make a decision). Thus,the lower and upper thresholds (π _(L), π _(U)) are computed accordingto Eq. 57:

$\begin{matrix}{{\underset{\_}{\pi}}_{L}\overset{\Delta}{=}\left\{ {{\begin{matrix}{{{{\hat{\sigma}}_{\eta}^{2}\lbrack n\rbrack}\left( {A - {Ce}^{({1 - n})}} \right)};} & {n \geq N_{\infty}} \\{{- \infty};} & {n < N_{\infty}}\end{matrix}\mspace{14mu}{and}{\overset{\_}{\pi}}_{U}}\overset{\Delta}{=}\left\{ {\begin{matrix}{{{{\hat{\sigma}}_{\eta}^{2}\lbrack n\rbrack}\left( {B + {Ce}^{({1 - n})}} \right)};} & {n \geq N_{\infty}} \\{{+ \infty};} & {n < N_{\infty}}\end{matrix},} \right.} \right.} & \left( {{Eq}.\mspace{14mu} 57} \right)\end{matrix}$

Another example of “gear-stepping” the upper and lower thresholds can beaccording to

${{\underset{\_}{\pi}}_{L} < {\overset{\sim}{S}}_{n} < {{\overset{\_}{\pi}}_{U} - \infty} < {\overset{\sim}{S}}_{n} < {+ \infty}};{n < N_{\infty}}$${{{{\hat{\sigma}}_{\eta}^{2}\lbrack n\rbrack}\left( {A - \frac{C}{\sqrt{n}}} \right)} < {\overset{\sim}{S}}_{n} < {{{\hat{\sigma}}_{\eta}^{2}\lbrack n\rbrack}\left( {B + \frac{D}{\sqrt{n}}} \right)}};{n \geq N_{\infty}}$where

$\begin{matrix}{{\underset{\_}{\pi}}_{L}\overset{\Delta}{=}\left\{ {{\begin{matrix}{{{{\hat{\sigma}}_{\eta}^{2}\lbrack n\rbrack}\left( {A - \frac{C}{\sqrt{n}}} \right)};} & {n \geq N_{\infty}} \\{{- \infty};} & {n < N_{\infty}}\end{matrix}\mspace{14mu}{and}{\overset{\_}{\pi}}_{U}}\overset{\Delta}{=}\left\{ {\begin{matrix}{{{{\hat{\sigma}}_{\eta}^{2}\lbrack n\rbrack}\left( {B + \frac{D}{\sqrt{n}}} \right)};} & {n \geq N_{\infty}} \\{{+ \infty};} & {n < N_{\infty}}\end{matrix},} \right.} \right.} & \left( {{Eq}.\mspace{14mu} 58} \right)\end{matrix}$are the corresponding lower and upper thresholds, respectfully.

Generalized-Likelihood-Ratio-Test (GLRT):

Use of the LLR and SPRT tests discussed above assume that certaincharacteristics of the container are known, such as the color of theinner wall of the container. In some cases, such characteristics of thecontainer may not be known. As a result, the parameter {circumflex over(μ)}₀[n] that represents an estimated mean of the null hypothesis y₀[n]signals cannot be derived, which would mean that the LLR and SPRT testscannot be used.

In such situations, the GLRT can be used instead, where the GLRT can usea different value to represent {circumflex over (μ)}⁰[n], as explainedbelow.

Let an observed received signal at time n be y[n] as follows:

$\begin{matrix}{{{{y\lbrack n\rbrack} = \begin{pmatrix}{y_{I}\lbrack n\rbrack} \\{y_{Q}\lbrack n\rbrack}\end{pmatrix}};{{d\lbrack n\rbrack} = {{r_{d}\lbrack n\rbrack}\begin{pmatrix}{\cos\;\theta_{d}} \\{\sin\;\theta_{d}}\end{pmatrix}}}},} & \left( {{Eq}.\mspace{14mu} 59} \right)\end{matrix}$where

$\begin{matrix}{{{\theta_{d} = {{\tan^{- 1}\frac{V_{c}}{U_{c}}} = {\tan^{- 1}\left( \frac{{T/2} - {2{{{T/4} - T_{c}}}}}{{T/2} - {2T_{c}}} \right)}}};}\left( {{for}\mspace{14mu}{square}\text{-}{wave}\mspace{14mu}{basis}\mspace{14mu}{functions}} \right)} & \left( {{Eq}.\mspace{14mu} 60} \right) \\{{\theta_{d} = \frac{4\pi\; L_{c}}{{Tc}_{m/s}}};\left( {{for}\mspace{14mu}{sinusoidal}\mspace{14mu}{basis}\mspace{14mu}{functions}} \right)} & \left( {{Eq}.\mspace{14mu} 61} \right)\end{matrix}$and d[n] is the point along the subspace defined by the phase θ_(d) inthe I/Q plane (an I/Q plane is shown in FIG. 6) that is r_(d)[n] fromthe origin where the round-trip ToF defines θ_(d) in radians for thegiven carrier period T and the speed of light c_(m/s). In Eq. 60, T_(c)is the round-trip time shift attributed to the container's reflection,and U_(c) and V_(c) are defined right above Eq. 16 further above. In Eq.61, L_(c) represents the length of the container.

In FIG. 6, the horizontal axis of the I/Q plane is the in-phase (I)axis, while the vertical axis is the quadrature-phase (Q) axis. Usingthe Cartesian distance for the norm, a vector y[n]−d[n] (represented byline 600 in FIG. 6) is the vector connecting the points y[n] and d[n]. Apoint 602 on the line 600 represents d[n]. The angle θ_(d) is betweenthe I axis and the line 600.

FIG. 6 further shows points 606 (blank circles) that are observedreceived signals y[n] corresponding to signals of a container includingat least one physical item. A darkened circle 608 is a mean (or otheraggregate) of the observed received signals y[n] corresponding tosignals of a container including at least one physical item. This meanis referred to as a “mean observation” in the ensuing discussion. Thedashed oval 610 around the line 600 represents the decision region ofthe GLRT. The origin endpoint 612 of the line 600 represents a containerwith a black wall, where no light is reflected. The opposite endpoint614 of the line 600 represents a container with a white wall, where alllight is reflected. Other points along the line 600 represent othercolors of the wall of the container.

The GLRT seeks to find the closest point along the line 600 to the meanobservation (608). In FIG. 6, such closest point is the point 602representing d[n]. This extrema point (602) is unique, so the GLRT cantake the derivative of the vector y[n]−d[n] with respect to r_(d)[n],

$\begin{matrix}{{{{y\lbrack n\rbrack} - {d\lbrack n\rbrack}}}_{2}^{2} = {{\begin{matrix}{{y_{I}\lbrack n\rbrack} - {{r_{d}\lbrack n\rbrack}\cos\;\theta_{d}}} \\{{y_{Q}\lbrack n\rbrack} - {{r_{d}\lbrack n\rbrack}\sin\;\theta_{d}}}\end{matrix}}_{2}^{2} = {\left( {{y_{I}\lbrack n\rbrack} - {{r_{d}\lbrack n\rbrack}\cos\;\theta_{d}}} \right)^{2} + \left( {{y_{Q}\lbrack n\rbrack} - {{r_{d}\lbrack n\rbrack}\sin\;\theta_{d}}} \right)^{2}}}} & \left( {{Eq}.\mspace{14mu} 62} \right) \\{{\frac{\partial}{\partial{r_{d}\lbrack n\rbrack}}{{{y\lbrack n\rbrack} - {d\lbrack n\rbrack}}}_{2}^{2}} = {{{\frac{\partial}{\partial{r_{d}\lbrack n\rbrack}}\left( {{y_{I}\lbrack n\rbrack} - {{r_{d}\lbrack n\rbrack}\cos\;\theta_{d}}} \right)^{2}} + {\frac{\partial}{\partial{r_{d}\lbrack n\rbrack}}\left( {{y_{Q}\lbrack n\rbrack} - {{r_{d}\lbrack n\rbrack}\sin\;\theta_{d}}} \right)^{2}\frac{\partial}{\partial{r_{d}\lbrack n\rbrack}}\left( {{y_{I}\lbrack n\rbrack} - {{r_{d}\lbrack n\rbrack}\cos\;\theta_{d}}} \right)^{2}}} = {{{{- 2}{y_{I}\lbrack n\rbrack}\cos\;\theta_{d}} + {2{r_{d}\lbrack n\rbrack}\left( {\cos\;\theta_{d}} \right)^{2}\frac{\partial}{\partial{r_{d}\lbrack n\rbrack}}\left( {{y_{Q}\lbrack n\rbrack} - {{r_{d}\lbrack n\rbrack}\sin\;\theta_{d}}} \right)^{2}}} = {{{- 2}{y_{Q}\lbrack n\rbrack}\sin\;\theta_{d}} + {2{r_{d}\lbrack n\rbrack}{\left( {\sin\;\theta_{d}} \right)^{2}.}}}}}} & \left( {{Eq}.\mspace{14mu} 63} \right)\end{matrix}$The derivative is set to zero, and the GLRT can solve for r_(d)[n] asfollows:

$\begin{matrix}{{{\frac{\partial}{\partial{r_{d}\lbrack n\rbrack}}{{{y\lbrack n\rbrack} - {d\lbrack n\rbrack}}}_{2}^{2}} = {{{{- 2}{y_{I}\lbrack n\rbrack}\cos\;\theta_{d}} + {2{r_{d}\lbrack n\rbrack}\left( {\cos\;\theta_{d}} \right)^{2}} - {2{y_{Q}\lbrack n\rbrack}\sin\;\theta_{d}} + {2{r_{d}\lbrack n\rbrack}\left( {\sin\;\theta_{d}} \right)^{2}}} = 0}},} & \left( {{Eq}.\mspace{14mu} 64} \right) \\{\mspace{20mu}{{{{r_{d}\lbrack n\rbrack}\left( {\left( {\cos\;\theta_{d}} \right)^{2} + \left( {\sin\;\theta_{d}} \right)^{2}} \right)} = {{{y_{I}\lbrack n\rbrack}\cos\;\theta_{d}} + {{y_{Q}\lbrack n\rbrack}\sin\;\theta_{d}}}},}} & \left( {{Eq}.\mspace{14mu} 65} \right) \\{\mspace{20mu}{{r_{d}\lbrack n\rbrack} = {{{y_{I}\lbrack n\rbrack}\cos\;\theta_{d}} + {{y_{Q}\lbrack n\rbrack}\sin\;{\theta_{d}.}}}}} & \left( {{Eq}.\mspace{14mu} 66} \right)\end{matrix}$

Note that r_(d)[n] may be negative depending on the combination ofobservation y[n] and the container's length L_(c). Because of thephysical interpretation, d[n] can be restricted to be the origin whenr_(d)[n] is negative, i.e.,

$\begin{matrix}{{\hat{d}\lbrack n\rbrack} = {{{{\hat{r}}_{d}\lbrack n\rbrack}\begin{pmatrix}{\cos\;\theta_{d}} \\{\sin\;\theta_{d}}\end{pmatrix}\mspace{14mu}{where}\mspace{14mu}{{\hat{r}}_{d}\lbrack n\rbrack}}\overset{\Delta}{=}\left\{ {\begin{matrix}{{r_{d}\lbrack n\rbrack};} & {0 \leq {r_{d}\lbrack n\rbrack}} \\{0;} & {0 > {r_{d}\lbrack n\rbrack}}\end{matrix}.} \right.}} & \left( {{Eq}.\mspace{14mu} 67} \right)\end{matrix}$

In accordance with some implementations of the present disclosure, theparameter {circumflex over (d)}[n] calculated above can be used as thebaseline vector {circumflex over (μ)}₀[n]. Given this estimate of{circumflex over (μ)}₀[n], either the LLR test or the SPRT can be thenused, as discussed above.

More generally, the GLRT involves defining a subspace (e.g. subspace 610in FIG. 6) based on a phase (e.g., θ_(d)), identifying a point (e.g.,d[n]) in the subspace that is closest to points representing measurementdata received from at least one sensor, using the identified point as anestimated baseline corresponding to an empty container, and computing avalue using the estimated baseline, where the value is computed by usingan LLR test or the SPRT.

In some examples as discussed above, the load status detector 208 andthe LRT function 209 (which can perform any of the LLR, SPRT, or GLRTtests, for example) can be implemented as machine-readable instructionsexecutable on one or more processors.

The machine-readable instructions can be stored in a non-transitorycomputer-readable or machine-readable storage medium. The storage mediumcan include one or multiple different forms of memory includingsemiconductor memory devices such as dynamic or static random accessmemories (DRAMs or SRAMs), erasable and programmable read-only memories(EPROMs), electrically erasable and programmable read-only memories(EEPROMs) and flash memories; magnetic disks such as fixed, floppy andremovable disks; other magnetic media including tape; optical media suchas compact disks (CDs) or digital video disks (DVDs); or other types ofstorage devices. Note that the instructions discussed above can beprovided on one computer-readable or machine-readable storage medium, oralternatively, can be provided on multiple computer-readable ormachine-readable storage media distributed in a large system havingpossibly plural nodes. Such computer-readable or machine-readablestorage medium or media is (are) considered to be part of an article (orarticle of manufacture). An article or article of manufacture can referto any manufactured single component or multiple components. The storagemedium or media can be located either in the machine running themachine-readable instructions, or located at a remote site from whichmachine-readable instructions can be downloaded over a network forexecution.

In the foregoing description, numerous details are set forth to providean understanding of the subject disclosed herein. However,implementations may be practiced without some of these details. Otherimplementations may include modifications and variations from thedetails discussed above. It is intended that the appended claims coversuch modifications and variations.

What is claimed is:
 1. A method performed by at least one processor,comprising: receiving measurement data from at least one sensor thatdetects a signal reflected from a surface inside a platform; applying alikelihood ratio test using the measurement data, wherein applying thelikelihood ratio test comprises: computing a plurality of values using alikelihood ratio test function applied on corresponding samples of themeasurement data received from the at least one sensor, and aggregatingthe plurality of values to produce an aggregate value; and determining aload status of the platform based on the aggregate value produced by thelikelihood ratio test.
 2. The method of claim 1, wherein applying thelikelihood ratio test comprises applying a log likelihood ratio test. 3.The method of claim 1, wherein determining the load status of theplatform is based on comparing the aggregate value to at least onethreshold, wherein the aggregate value having a first relationship tothe at least one threshold indicates that the platform is empty, and theaggregate value having a second relationship to the at least onethreshold indicates that the platform is loaded.
 4. The method of claim3, further comprising: computing the at least one threshold based ontuning parameters selected to control error rates.
 5. The method ofclaim 1, wherein applying the likelihood ratio test comprises applying ageneralized likelihood ratio test.
 6. A method performed by at least oneprocessor, comprising: receiving measurement data from at least onesensor that detects a signal reflected from a surface inside a platform;applying a generalized likelihood ratio test using the measurement data,wherein applying the generalized likelihood ratio test comprises:defining a subspace based on a phase, identifying a point in thesubspace that is closest to points representing the measurement datareceived from the at least one sensor, using the identified point as anestimated baseline corresponding to an empty platform; and computing avalue using the estimated baseline; and determining a load status of theplatform based on the generalized likelihood ratio test.
 7. The methodof claim 6, wherein computing the value using the estimated baselinecomprises computing the value using a log-likelihood ratio test or asequential probability ratio test.
 8. An apparatus comprising: at leastone sensor to output measurement data responsive to detecting a signalreflected from a surface inside a platform; and at least one processorconfigured to: compute a value by applying a likelihood ratio testfunction on the measurement data, wherein the likelihood ratio testfunction is based on a ratio between a model representing an emptyplatform and a model representing a loaded platform, and determine aload status of the platform based on the computed value.
 9. Theapparatus of claim 8, wherein the at least one sensor comprises a lightdetector to detect reflected light emitted by a light emitter.
 10. Theapparatus of claim 8, wherein the model representing the empty platformis based on a mean of measurement data samples representing the emptyplatform, and the model representing the loaded platform is based on amean of measurement data samples representing the loaded platform. 11.The apparatus of claim 8, wherein the at least one processor isconfigured to: compute a plurality of values output by applying thelikelihood ratio test function on a plurality of measurement datasamples from the at least one sensor; and aggregate the plurality ofvalues to produce an aggregate value, wherein the determining of theload status is based on comparing the aggregate value to at least onethreshold.
 12. The apparatus of claim 8, wherein the at least oneprocessor is configured to compare the computed value to at least onethreshold, wherein the computed value having a first relationship to theat least one threshold indicates that the platform is empty, and thecomputed value having a second relationship to the at least onethreshold indicates that the platform is loaded.
 13. The apparatus ofclaim 8, wherein the at least one processor is configured to: determinean estimated baseline corresponding to an empty platform by: identifyinga point in a subspace that is closest to points representing measurementdata received from the at least one sensor, wherein the subspacecorresponds to a specified phase in In-phase/Quadrature (IQ) space,wherein the likelihood ratio test function is based on the estimatedbaseline.
 14. A non-transitory machine-readable storage medium storinginstructions that upon execution cause a system to: receive measurementdata from at least one sensor that detects a signal reflected from asurface inside a platform; apply a likelihood ratio test using themeasurement data, the likelihood ratio test based on a ratio between amodel representing an empty platform and a model representing a loadedplatform, and the likelihood ratio test producing an output value; anddetermine a load status of the platform based on comparing the outputvalue to at least one threshold.
 15. The non-transitory machine-readablestorage medium of claim 14, wherein the determining comprises:indicating that the platform is empty responsive to the output valuehaving a first relationship with respect to the at least one threshold,and indicating that the platform is loaded responsive to the outputvalue having a second relationship with respect to the at least onethreshold.
 16. The non-transitory machine-readable storage medium ofclaim 14, wherein the at least one sensor comprises a light detector todetect reflected light emitted by a light emitter.
 17. Thenon-transitory machine-readable storage medium of claim 14, wherein themodel representing the empty platform is based on a mean of measurementdata samples representing the empty platform, and the model representingthe loaded platform is based on a mean of measurement data samplesrepresenting the loaded platform.
 18. The non-transitorymachine-readable storage medium of claim 14, wherein the instructionsupon execution cause the system to: compute a plurality of values outputby applying a likelihood ratio test function of the likelihood ratiotest on a plurality of measurement data samples from the at least onesensor; and aggregate the plurality of values to produce an aggregatevalue, wherein the determining of the load status is based on comparingthe aggregate value to the at least one threshold.
 19. Thenon-transitory machine-readable storage medium of claim 14, wherein theinstructions upon execution cause the system to: compute the at leastone threshold based on tuning parameters selected to control errorrates.